Is there any way to see on the UI which weapons are damaged, if for example you have a broadside weapon, and one side is offline? It seems like when I hover the weapon I don't get any new information about that, and I have to look at the ship model carefully in space to see which of the sides is actually damaged. Perhaps that wasn't a destroyed weapon, and if the weapon were fully destroyed one of the broadside firing arcs would go out? I'm just trying to learn how to determine which side to face when weapons are damage or destroyed as quickly as possible, since it's possible for your whole ship to be obscured by particle effects like the Dark Eldar AoE (Dark Matter maybe?).

If you double tap alt to get the extra ui details one of the things it adds is little colored pins of light that come up from damaged components to make them easier to spot at a glance, as well as highlighting of damaged model bits on some weapons.

It's something I wish was default but it is in there. You can also pull it up momentarily by holding alt, but I prefer the always on double-tap.

As far as I know there's no menu to show it when the model is obscured.

Apart from the above, if you open the details panel of a ship (bottom left side), you see icons for all weapons, and below them the current/basic number (so 1/3 Lance Turrets would mean 2 are destroyed). Also mouseovering the icons will show you current range.